On weak convergence of entropy solutions to scalar conservation laws

We prove that weak limits of entropy solutions to a one-dimensional scalar conservation law are entropy solutions as well. We consider a scalar conservation law ut + f(u)x = 0, (t, x) ∈ Π = (0, +∞)× R. (1) The flux function f(u) is supposed to be only continuous: f(u) ∈ C(R). Recall the notion of an entropy solution of (1) in the sense of Kruzhkov [6]. Definition 1. A bounded measurable function u = u(t, x) ∈ L∞(Π) is called an entropy solution (e.s. for short) of (1) if ∀k ∈ R ∂ ∂t |u− k|+ ∂ ∂x [sign(u− k)(f(u)− f(k))] ≤ 0 (2) in the sense of distributions on Π ( in D′(Π) ). Here sign u = { 1 , u > 0, −1 , u ≤ 0. , and relation (2) means that for each test function h = h(t, x) ∈ C 0(Π), h ≥ 0 ∫ Π [|u− k|ht + sign(u− k)(f(u)− f(k))hx]dtdx ≥ 0. Taking in (2) k = ±R, R ≥ ‖u‖∞, we derive that ut + f(u)x = 0 in D′(Π), i.e. an e.s. u = u(t, x) is a weak solution of (1). We recall also that u = u(t, x) is an e.s. of the Cauchy problem for (1) with initial data